07 - Polynomials on Matrices

Properties of Polynomials on Matrices

Let $A\in M_n(\mathbf{F})$ and $f(x)=a_k{x}^k+\cdots +a_0\in \mathbf{F}[x]$ then

$$f(A):=a_k{A}_k+\cdots +a_{0}I\in M_n(\mathbf{F})$$

As $A^p{A}^q=A^q{A}^p$ and $\lambda A=A\lambda$ for all $p,q\ge 0$ and $\lambda \in \mathbf{F}$ then
For all $f(x),g(x)\in \mathbf{F}[x]$ then

$$\begin{array}{rl}& f(A)g(A)=g(A)f(A)\\ & Av=\lambda v⟹f(A)v=f(\lambda )v\end{array}$$

Note that these can be deduced by looking at ring homomorphism $E_A:\mathbb{F}[x]\to M_n(\mathbb{F})$

Annihilating Polynomial of a Matrix lemma

For all $A\in M_n(\mathbf{F})$ there exists a non-zero polynomial $f(x)\in \mathbf{F}[x]$ such that

$$f(A)=0$$

Proof

Note the dimension $\dim M_n(\mathbf{F})=n\times n$ is finite
Hence

$$\{I,A,A^2,\cdots ,A^k\}\subseteq M_n(\mathbf{F})\text{ is linearly dependent for }k\ge n^2$$

So by definition there exist scalars $a_i\in \mathbf{F}$ (not all $0$) such that

$$a_k{A}^k+\cdots +a_{0}I=0$$

Hence

$$f(x)=a_k{x}^k+\cdots +a_0$$

is an annihilating polynomial

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Minimal Polynomial of $A$

Defined as the the monic polynomial $p(x)$ of least degree such that $p(A)=0$
Commonly written as

$$m_A(x)$$

Characteristic Polynomial of $A$

Defined as

$${\chi }_A(x)=\det (A-xI)$$

Alternate form of Characteristic Polynomial of $A$ lemma

$${\chi }_A(x)=(-1{)}^n{x}^n+(-1{)}^{n-1}\mathrm{Tr}(A)x^{n-1}+\cdots +\det A$$

Polynomials are similar under similar matrices

For $A,C\in M_n(\mathbf{F})$ similar so there exists $P\in M_n(\mathbf{F})$ such that $C=P^{-1}AP$
Then for any polynomial $f(X):=a_k{X}_k+\cdots +a_{0}I\in M_n(\mathbf{F})$

$$f(C)=f(P^{-1}AP)=P^{-1}f(A)P$$

Proof $k\ge 0$ then

For

$$C^k=(P^{-1}AP{)}^k=\underset{k\text{ times}}{\underbrace{(P^{-1}AP)(P^{-1}AP)\cdots (P^{-1}AP)}}=P^{-1}{A}^{k}P$$

as $P{P}^{-1}=I$ when looking at the middle terms

So

$$\begin{array}{rl}f(C)& =a_k{C}^k+\cdots +a_{0}I\\ & =a_k{P}^{-1}{A}^{k}P+\cdots +a_0{P}^{-1}IP\\ & =P^{-1}(a_k{A}^k+\cdots +a_{0}I)P\\ & =P^{-1}f(A)P\end{array}$$

Minimal Polynomial is invariant for similar matrices

Let $C,P,A$ be $(n\times n)-$matrices such that $C=P^{-1}AP$ so ($C$ is similar to $A$) then

$$m_C(x)=m_A(x)$$

Proof

As we know polynomials are similar applied to similar matrices then

$$f(C)=f(P^{-1}AP)=P^{-1}f(A)P$$

for all polynomials $f$

Hence

$$0=m_C(C)=P^{-1}{m}_C(A)P\text{ and so }{m}_C(A)=0$$

As $m_A\mid m_C$ and similarly $m_C\mid m_A$ then $m_C=m_A$ as both are monic

Characteristic Polynomial is invariant under similar matrices

Let $C,P,A$ be $(n\times n)-$matrices such that $C=P^{-1}AP$ so ($C$ is similar to $A$) then

$${\chi }_C(x)={\chi }_A(x)$$

Proof

$$\begin{array}{rl}{\chi }_C(x)& =\det (C-xI)\\ & =\det (P^{-1}AP-xI)\\ & =\det (P^{-1}AP-P^{-1}\cdot xI\cdot P)\\ & =\det (P^{-1}(A-xI)P)\\ & =\det (P^{-1})\cdot \det (A-xI)\cdot \det P\\ & =\det (A-xI)\text{ as }\det P\cdot \det P^{-1}=1\\ & ={\chi }_A(x)\end{array}$$

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Upper Triangularisation of a Matrix corollary

Let $A$ be a $n\times n$ matrix
If ${\chi }_A(x)$ is a product of linear factors then there

$$\exists P\in M_n(\mathbf{F})\text{ such that }{P}^{-1}AP\text{ is upper triangular}$$

Annihilating Polynomial of an Upper Triangular Matrix

Let $A$ be an upper triangular $(n\times n)-$matrix with diagonal entries ${\lambda }_1,\cdots ,{\lambda }_n$ then

$$\prod _{i=1}^n(A-{\lambda }_{i}I)=0$$

Proof

Let $e_1,\cdots ,e_n$ be the standard basis vectors for ${\mathbf{F}}^n$ then

$$(A-{\lambda }_{n}I)v\in ⟨e_1,\cdots ,e_{n-1}⟩\text{ for all }v\in {\mathbf{F}}^n$$

So generally

$$(A-{\lambda }_{i}I)w\in ⟨e_1,\cdots ,e_{i-1}⟩\text{ for all }w\in ⟨e_1,\cdots ,e_i⟩$$

Since

$$\begin{array}{rl}\mathrm{Im}(A-{\lambda }_{n}I)& \subseteq ⟨e_1,\cdots ,e_{n-1}⟩\\ \mathrm{Im}(A-{\lambda }_{n-1}I)(A-{\lambda }_{n}I)& \subseteq ⟨e_1,\cdots ,e_{n-2}⟩\end{array}$$

Then we get

$$\prod _{i=1}^n(A-{\lambda }_{i}I)=0$$

Triangularisability Criterion

$$\begin{array}{rl}T& \text{ is triangularisable (over a given field)}\\ & ⟺{\chi }_T\text{ factors as a product of linear polynomials (over that field)}\\ & ⟺\text{each }{f}_i\text{ is linear}\\ & ⟺m_T\text{ factors as a product of linear polynomials}\end{array}$$